ArticlePDF Available

## Abstract

It is pointed out that a complete resolution of the twin paradox demands that the travelling twin takes into account the gravitational effect upon the rate of time when he predicts the ageing of his brother. Two ways of making this prediction are presented. The first one is formally very simple. However, it involves an assumption that is not obvious when the travelling twin stipulates the distance of his brother. The second method makes use of Lagrangian dynamics in a uniformly accelerated reference frame. Then no such assumption is necessary.
INSTITUTE OF PHYSICS PUBLISHING EUROPEAN JOURNAL OF PHYSICS
Eur. J. Phys. 27 (2006) 885–889 doi:10.1088/0143-0807/27/4/019
The twin paradox in the theory of
relativity
ØGrøn
Oslo College, Department of Engineering, PO Box 4, St Olavs Pl., 0130 Oslo, Norway
and
Institute of Physics, University of Oslo, PO Box 1048 Blindern, 0216 Oslo, Norway
Published 30 May 2006
Online at stacks.iop.org/EJP/27/885
Abstract
It is pointed out that a complete resolution of the twin paradox demands that the
travelling twin takes into account the gravitational effect upon the rate of time
when he predicts the ageing of his brother. Two ways of making this prediction
are presented. The ﬁrst one is formally very simple. However, it involves an
assumption that is not obvious when the travelling twin stipulates the distance
of his brother. The second method makes use of Lagrangian dynamics in a
uniformly accelerated reference frame. Then no such assumption is necessary.
1. Introduction
There exist more than three hundred articles about the twin paradox . In the present
paper, I shall present two ways that the travelling twin can predict the ageing of his brother. As
far as I have found, none of these calculations have been presented before. They are interesting
because they are simple and exact. In both calculations, one has to take into account the effect
of gravity upon time. Hence, a complete resolution of the twin paradox involves the general
theory of relativity.
This is not so surprising. The formulation of the twin paradox makes use of the general
principle of relativity, which is not contained in the special theory of relativity. One says:
One twin, say Eartha, at rest on the Earth, predicts that her sister, Stella, is younger than
herself when they meet again after Stella’s travel, because of the kinematic time dilation. But
according to the general principle of relativity, Stella can consider herself as at rest during all
of the travel, and would then predict that Eartha is younger than herself when they meet again.
The twin paradox is the conﬂict between these predictions.
Hence, the twin paradox arises from using general relativity in its formulation and only
special relativity to calculate the ageing that each twin predicts for the other. In the present
paper, it will be shown how general relativity is involved in the solution of the twin paradox.
0143-0807/06/040885+05\$30.00 c
2006 IOP Publishing Ltd Printed in the UK 885
886 ØGrøn
2. Elementary solution of the twin paradox
The star Alpha Proxima is L0=4 ly (light years) from the Sun. Stella travels with constant
velocity v=0.8cto Alpha Proxima and back. To simplify the calculation we assume
that the acceleration at the start, the turning point and the arrival is so large that these
brief periods of acceleration can be neglected. As measured by Eartha the travel time is
t =2L0/v =8ly/0.8c=10 years. Hence, Eartha predicts that she will be 10 years older
when she meets Stella after her travel, and that Stella will only be
τS=1v2/c2(2L0/v) =1(0.8c)2/c2·10 years =6 years (2.1)
older. According to Eartha, Stella has become 4 years younger than herself during the travel
due to the velocity-dependent time dilation.
Let us now see what Stella predicts. The general principle of relativity says that she
can consider herself as at rest and Eartha as the traveller during all of the time when they
are separated from each other. She observes that the Earth and Alpha Proxima travel with
the velocity v=0.8c. Then the distance between the Earth and Alpha Proxima is Lorentz
contracted, so the distance is L=L01v2/c2=4ly0.6=2.4 ly. Stella calculates that
the time Eartha uses forth and back is 2(L/v) =2(2.4/0.8)years =6 years. So Stella
predicts that she ages by 6 years during Eartha’s travel. This agrees with Eartha’s prediction.
Stella’s prediction of Eartha’s ageing remains. Due to the velocity-dependent time dilation
she ﬁnds that Eartha ages by 1v2/c2(2L/v) =0.6·6 years =3.6 years during her forth
and back travel, in contradiction to Eartha’s own prediction that she ages by 10 years. This
disagreement is just the twin paradox.
What has not yet been taken into account is the effect upon time of the gravitational ﬁeld
that Stella experiences at the start, the turning point and the arrival. We assume that Stella
has constant proper acceleration, g, i.e. constant acceleration relative to her instantaneous
inertial rest frame, during these periods. Hence, she may be considered at rest in a uniformly
accelerated reference frame. In such a frame the line-element has the form
ds2=−
1+ gx
c22
c2dt2+dx2+dy2+dz2.(2.2)
The general physical interpretation of the line element in the case of a timelike spacetime
interval is that it represents the proper time interval dτas measured by a clock following a
worldline connecting two nearby events with coordinate separations (dt,dx, dy, dz), i.e.
ds2=−c2dτ2.(2.3)
For a clock at rest at the position x=hthis gives
τ =1+ gh
c2t. (2.4)
Stella chooses the coordinate system so that she is at the origin each time she accelerates,
i.e. each time she experiences a gravitational ﬁeld. At the start and the arrival Eartha is at
a small distance from her. In the limit of inﬁnite acceleration corresponding to vanishingly
short periods of acceleration, the periods with gravity at the start and arrival can be neglected.
Consider the start, for example. The fact that gh is ﬁnite in the limit g→∞is most
simply seen from the Galilean equation v2=2gh since the velocity is ﬁnite. The relativistic
formulae are more complicated, but lead to the same result. Furthermore tstart =v/g
vanishes as g→∞. Hence τstart vanishes at the start in this limit, and likewise at the arrival.
Note that this limit must be taken by Stella in order that she shall consider the same situation
The twin paradox in the theory of relativity 887
as Eartha, who reckoned with constant velocities all the way, neglecting the periods when
Let us now see how Stella can predict the ageing of Eartha when she experiences a
gravitational ﬁeld at the turning point close to Alpha Proxima. The Earth with Eartha falls
freely upwards in the gravitational ﬁeld she experiences. When the Earth stops and begins
to fall down it is at a height h=4 ly. Stella experiences the gravitational ﬁeld for a time
t =2v/g. Hence, Stella calculates that Eartha ages by
τ =1+ gh
c2t =1+ gh
c22v
g
=
2v
g+2hv
c2.(2.5)
The ﬁrst term vanishes in the limit g→∞. In this limit, Stella ﬁnds that Eartha ages by
τ =
2hv
c2=
2·4ly0.8c
c2=6.4 years (2.6)
during the time she changes from moving freely upwards with velocity vto falling downwards
with the same velocity.
Stella found that Eartha aged by 3.6 years during the forth and back travel. Hence the total
ageing of Eartha according to Stella is 10 years in agreement with Eartha’s own prediction.
3. Using relativistic Lagrangian dynamics in solving the twin paradox
We shall now see how Stella can calculate the ageing of Eartha during the time she experiences
a gravitational ﬁeld when she is close to Alpha Proxima, without making the assumption that
the distance of Eartha from her can be put equal to 4 ly during the whole of this period.
Eartha is moving freely in the gravitational ﬁeld experienced by Stella. The covariant
Lagrangian of Eartha, considered as a free particle with unit mass, is
L=1
2gµν ˙
xµ˙
xν(3.1)
where a dot denotes differentiation with respect to Eartha’s proper time. In the present case,
L=−
1
21+ gx
c22
c2˙
t2+1
2˙
x2.(3.2)
Since the Lagrangian is independent of the time, the canonical momentum
pt=−
1+ gx
c22
c˙
t(3.3)
is a constant of motion. The four-velocity identity takes the form
1+ gx
c22
c2˙
t2+˙
x2=−c2.(3.4)
Inserting the expression for ptinto the four-velocity identity gives
p2
tc21+ gx
c22
=1+ gx
c22˙
x2(3.5)
or
dτ=
1+ gx
c2
p2
tc21+ gx
c22dx. (3.6)
888 ØGrøn
Eartha’s turning point is at x=h. Hence, the value of ptmay be determined from the
condition ˙
x=0forx=h. Using equation (3.4)thisgives
pt=c1+ gh
c2.(3.7)
Let x1be Eartha’s position in Stella’s coordinate system at the moment when Stella ﬁrst
experiences the gravitational ﬁeld. At this moment the Earth and Alpha Proxima move upwards
with a velocity vand a Lorentz contracted distance, x1=2.4 ly. During the slowing down
the distance of Eartha increases from x1=2.4lyto x2=h=4 ly. Integrating equation (3.5)
from x1to x2and inserting the value of pt, Stella ﬁnds that during this slowing down Eartha
ages by
τ12=
c
g1+ gx2
2
c22
1+ gx2
1
c22
.(3.8)
Taking the limit with inﬁnite proper acceleration, Stella obtains
lim
g→∞
τ12=
1
cx2
2x2
1.(3.9)
Inserting x2=4 ly and x1=2.4 ly. Stella ﬁnds limg→∞ τ12=3.2 years. So Stella ﬁnds that
Eartha’s ageing as she changes her motion from upwards with velocity v=0.8cto downwards
with the same velocity is τEartha =2 limg→∞ τ12=6.4 years. Hence, Stella predicts a total
ageing of Eartha, 3.6 years + 6.4 years =10 years, again in agreement with the prediction
of Eartha herself.
4. Conclusion
The formulation of the twin paradox involves the general theory of relativity by making use
of the general principle of relativity. It arises when the ‘travelling’ twin considers herself as at
rest during all parts of the travel, including the periods when she is accelerated, and predicts
that the Earthbound twin is younger than herself when they meet after the departure, making
use of the special relativistic kinematic time dilation, only.
Its solution is to recognize that the travelling twin must take the effect of gravity upon
time into account when she calculates the age of her sister. When the ‘travelling’ twin turns
around at Alpha Proxima she experiences a gravitational ﬁeld directed away from her sister.
Hence, the Earthbound sister is higher up in this ﬁeld than herself, and ages faster. This effect
is greater the stronger the gravitational ﬁeld she experiences, so it cannot be neglected even in
the limit of a vanishing short period of acceleration, because the proper acceleration increases
towards inﬁnity in this limit.
Although Eartha and Stella both consider themselves as at rest, they both predict that
Eartha ages by 10 years during their departure and Stella only 6 years. But they have totally
different explanations for this difference of age when they meet again. Eartha says that Stella
is younger than herself because of Stella’s slower ageing due to her great velocity. Stella,
on the other hand, says that Eartha is older than herself because she ages so fast when Stella
experiences a gravitational ﬁeld close to Alpha Proxima. This illustrates a general property of
physical theories. The explanation that an observer gives of a phenomenon depends upon the
motion of the observer.
The twin paradox in the theory of relativity 889
References
 Marder L 1971 Time and the Space-Traveller (London: George Allen & Unwin)
 Møller C 1972 The Theory of Relativity 2nd edn (Oxford: Oxford University Press) section 8.17
 Fock V 1966 The Theory of Space, Time and Gravitation 2nd edn (New York: Pergamon) section 62
 Unruh W G 1981 Parallax distance, time, and the twin ‘paradox’ Am.J.Phys.49 589–92
 Eriksen E and Grøn Ø 1990 Relativistic dynamics in uniformly accelerated reference frames with application to
the clock paradox Eur. J. Phys. 11 39–44
 Debs T A and Redhead M L G 1995 The twin ‘paradox’ and the conventionality of simultaneity Am. J. Phys.
64 384–92
 Nicoli´
c H 2000 The role of acceleration and locality in the twin paradox Found. Phys. Lett. 13 595–601
 Iorio L 2006 An analytical treatment of the Clock Paradox in the framework of the special and general theories
of relativity Found. Phys. Lett. 18 1–19